On the reconstruction of the covariance of stationary Gaussian processes observed through zero-memory nonlinearities--Part II (Corresp.)

The problem of reconstructing the varianceR(0)of a zero-mean stationary Gaussian process passed through a memoryless nonlinearityf(x)is considered whenfand the first two moments of the output     

The prediction error of stationary Gaussian time series of unknown covariance

In prediction problems of communication and control theory, it has become increasingly obvious that there are many applications in which a priori assumptions regarding data statistics are not justified. Thus, systems must be designed to take maximum advantage of whatever statistics are encountered. Unfortunately, these systems are inherently nonlinear in operation, which makes it difficult, ff not impossible, to evaluate their performance. In this paper the asymptotic form of the mean square prediction error is found for a stationary Gaussian time series when the prediction is a linear weighting of the immediate past, the weights being "learned" from the data. Computer results are given to demonstrate the usefulness of the asymptotic formula.     

On the covariance function of stationary binary sequences withgiven mean

Recent work on characterizing the covariance functions of stationary binary sequences has been geometric in flavor. Extreme distributions induced in windows of a given length are obtained from special periodic sequences. The work is extended to sequences with a given mean m, revealing some finer structure     

Parameter estimation of continuous-time stationary Gaussian processes with rational spectra

This paper considers the problem of estimating the parameters of continuous-time stationary Gaussian processes with rational spectra, from uniformly sampled measurements. The sampled process is shown to be an autoregressive moving-average process, and explicit relationships between the parameters of the continuous-time and the sampled processes are derived. These relationships are then used to derive a lower bound on the variances of biased estimates of the continuous-time parameters, and on the generalized variance of such estimates. It is shown by some examples that the bound on the generalized variance depends on the sampling interval in a nonmonotonic manner. In particular, for each specific set of parameters there exists a sampling interval for which the lower bound is minimized.This work was supported by the Army Research Office under Contract Number DAAG29-83-C-0027.     

On the asymptotic capacity of stationary Gaussian fading channels

We consider a peak-power-limited single-antenna flat complex-Gaussian fading channel where the receiver and transmitter, while fully cognizant of the distribution of the fading process, have no knowledge of its realization. Upper and lower bounds on channel capacity are derived, with special emphasis on tightness in the high signal-to-noise ratio (SNR) regime. Necessary and sufficient conditions (in terms of the autocorrelation of the fading process) are derived for capacity to grow double-logarithmically in the SNR. For cases in which capacity increases logarithmically in the SNR, we provide an expression for the "pre-log", i.e., for the asymptotic ratio between channel capacity and the logarithm of the SNR. This ratio is given by the Lebesgue measure of the set of harmonics where the spectral density of the fading process is zero. We finally demonstrate that the asymptotic dependence of channel capacity on the SNR need not be limited to logarithmic or double-logarithmic behaviors. We exhibit power spectra for which capacity grows as a fractional power of the logarithm of the SNR     

Synthesis of multivariate Gaussian random processes with a preassigned covariance (Corresp.)

Evaluation of complex systems in a laboratory environment requires the generation of inputs to the system sensors that are representative of the operational environment. It is therefore necessary to synthesize input test signals that reflect the mutual dependencies found in situ. For multivariate Gaussian inputs, algorithms are derived allowing 1) the transformation of dependent Gaussian random variables into independent variables; 2) the generation of jointly Gaussian random variables with a constant covariance matrix; and 3) the synthesis of stationary multivariate Gaussian random processes. These algorithms have simple electronic hardware and computer software implementations that will facilitate the laboratory evaluation and digital computer simulation of complex systems.     

Representation of stationary Gaussian processes on a finite interval (Corresp.)

A stationary Gaussian processxi_{t}, with a spectral densityfsatisfyingint (log f)(1 + x_{2})^{-1} dx > inftyagrees in a finite interval [- T,T] with a process et which has a purely discrete spectral measure.     

Correlation matrix for quadrature components of two cross-correlated stationary narrow-band Gaussian processes

Collecting the work of several previous authors concerning the cross-correlation functions of stationary narrow-band Gaussian processes with their Hilbert transforms, a brief derivation is given for the correlation matrix for the quadrature components of two cross-correlated stationary narrow-band Gaussian processes. The elements of the matrix are represented in terms of the auto-correlation functions RN(τ) and RN2(τ) of the two Gaussian processes, the cross-correlation function R12(τ), and the Hilbert transforms of these functions. Alternative representations are given in terms of the power density spectra S1(ω) and S2(ω) and the cross density spectrum S12(ω) of the two processes. The sixteen matrix elements are found to consist of a maximum of six independent functions.     

On the detection of known binary signals in Gaussian noise of exponential covariance

The optimum test statistic for the detection of binary sure signals in stationary Gaussian noise takes a particularly simple form, that of a correlation integral, when the solution, denoted byq(t), of a given integral equation is well behaved(L_{2}). For the case of a rational noise spectrum, a solution of the integral equation can always be obtained if delta functions are admitted. However, it cannot be argued that the test statistic obtained by formally correlating the receiver input with aq(t)which is notL_{2}is optimum. In this paper, a rigorous derivation of the optimum test statistic for the case of exponentially correlated Gaussian noiseR(tau) = sigma^{2} e^{-alpha|tau|}is obtained. It is proved that for the correlation integral solution to yield the optimum test statistic whenq(t)is notL_{2}, it is sufficient that the binary signals have continuous third derivatives. Consideration is then given to the case where a, the bandwidth parameter of the exponentially correlated noise, is described statistically. The test statistic which is optimum in the Neyman-Pearson sense is formulated. Except for the fact that the receiver employsalpha_{infty}(which in general depends on the observed sample function) in place ofalpha, the operations of the optimum detector are unchanged by the uncertainty inalpha. It is then shown that almost all sample functions can be used to yield a perfect estimate ofalpha. Using this estimate ofalpha, a test statistic equivalent to the Neyman-Pearson statistic is obtained.     

Evolution of the probability density functions of Gaussian ASE noise in zero-memory nonlinear fiber

The propagation of Gaussian amplified emission noise in nonlinear optical fiber with negligible dispersion is considered. It is well known that fiber Kerr nonlinearity causes a nonlinear phase noise, also known as the Gordon–Mollenauer effect. In this paper, we examine the effect of the Kerr nonlinearity on the probability density functions of the Gaussian noise quadratures. This can lead to large deviations from the Gaussian statistics of the amplified spontaneous emission noise. Analytical statistics of the noise quadratures at the output of nonlinear fiber with negligible dispersion are derived and compared with numerical simulations. The statistics of the noise squared envelope are also presented and, based on these results, the influence of Kerr nonlinearity on the direct detection and coherent detection amplitude modulated links is discussed.     

On Walsh differentiable dyadically stationary random processes

Some basic properties of dyadically stationary (DS) processes are introduced, including continuity and spectral representation. A sampling theorem based on the Walsh functions is investigated for random signals that are not necessarily sequency-limited. By using the concept of a dyadic derivative, the resulting aliasing error is calculated together with the speed of convergence. An example gives a glimpse into the possibilities of applying the sampling theorem as well as the dyadic derivative.     

Zero-mean proper complex Gaussian random processes through nonlinear channels

The statistical characterisation of the output of a memory less nonlinear device having, as an input, a zero mean proper complex Gaussian random process is studied. The closed form expression obtained for the high-order moments involved allowed for neat mathematical expressions for the autocorrelation function and the power spectral density of the signal at the nonlinearity output. Parts of the autocorrelation function and of the power spectral density that correspond to the desired signal and to the intermodulation products of different orders are easily identified.     

Information capacity of the stationary Gaussian channel

The averaged information capacity of the mismatched stationary continuous-time Gaussian channel is considered, in the limit as the observation time becomes infinite. It is required only that the signal satisfy a reproducing kernel Hilbert space constraint on expected energy. This requires the, signal energy to be distributed in regions where the noise energy, is not too small, and is the weakest condition that provides finite capacity. In the case when both the noise covariance and the filter on the message are expressed by rational functions, the results complement those previously obtained by Gallager and Holsinger (1968), and the combination gives a complete solution to the capacity problem. The treatment provides a desirable generality on the class of transmitted signals that is not present in previous treatments     

On the approximation of the discrete Karhunen-Loeve transform for stationary processes

Suboptimal fast transforms are useful substitutes to the optimal Karhunen-Loève transform (KLT). The selection of an efficient approximation for the KLT must be done with respect to some performance criterion that might differ from one application to another. A general class of criterion functions including most of the commonly used performance measures is introduced. They are shown to be optimized by the KLT. Various properties of the eigenvectors of the symmetric Toeplitz covariance matrix of a wide sense stationary process are reviewed. Several transforms such as the complex or real, odd and even Fourier transforms (DFT, DOFT, DREFT, DROFT), the cosine and even sine transforms (DCT, DEST) are obtained from the decomposition of a symmetric Toeplitz matrix in the sum of a circulant and a skew circulant matrix. These transforms are compared on the basis of a general performance criterion and appear to be good substitutes for the optimal KLT. Finally, it is shown that these transforms are asymptotically equivalent in performances to the KLT of an arbitrary wide sense stationary process.     

On the error matrix in optimal linear filtering of stationary processes

The error covariance matrix corresponding to optimal linear causal filtering of second-order stationary processes in additive noise is considered. Formulas expressing this error matrix in terms of the optimal transfer function are established, and in the nonsingular case the optimal transfer function is expressed in terms of the spectral densities. These are straightforward generalizations of previously published scalar results, and the derivation is similarly based on Hardy space theory. Explicit bounds on the minimal error (i.e., the trace of the optimal error covariance matrix) are obtained for filtering in white noise. Furthermore, an explicit expression for the error covariance matrix is derived for the case of transmitting the same signal over several white-noise channels.     

Output correlation functions of power-law nonlinearities fed by Gaussian noise

Some of Middleton's results concerning halfwave and fullwave rectifiers fed by (narrowband) noise are simplified and broadened in their application. Examples, including the hard limiter, are considered, and general results are obtained for the shapes of the tails of the output spectra.     

Interpolation of complex stationary processes

The problem of minimum mean-square infinite extent interpolation for discrete-time stationary complex stochastic processes is studied. The interpolator consists of linear combinations of samples of the process and of their complex conjugate. The expressions of the interpolator and of the approximation error are derived and various consequences are examined. It is shown in particular that the approximation error may be zero while the interpolation error obtained when using only linear combinations of the samples is maximum